- Substitute \(y = 64\) into the equation \(y = \frac{1}{2}x^2\) to obtain \(64 = \frac{1}{2}x^2\).
- Multiply by 2 to find \(x^2 = 128\).
- Solve for \(x\) by taking the square root to get \(x = \pm 8\sqrt{2}\).
- Identify the solutions as \((8\sqrt{2}, 64)\) and \((-8\sqrt{2}, 64)\), with the positive side length being \(\boxed{8\sqrt{2}}\) feet.
Explanation
- Analysis of the System
The system of equations is given by:
\[ \begin{cases} y = \frac{1}{2} x^2 \ y = 64 \end{cases} \]
We are looking for the points \((x, y)\) that satisfy both equations simultaneously. Since the problem involves a physical context—the side length of a sun shade—we are also interested in the positive value of \(x\). We will first solve for the variable \(x\) by substituting the value of \(y\) from the second equation into the first.
- Solving for x
Substituting \(y = 64\) into the equation \(y = \frac{1}{2} x^2\), we get:
\[64 = \frac{1}{2} x^2\]
To isolate \(x^2\), multiply both sides by 2:
\[128 = x^2\]
Taking the square root of both sides, we find:
\[x = \pm \sqrt{128}\]
Simplifying the square root, \(\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}\). Therefore, the potential values for \(x\) are \(8\sqrt{2}\) and \(-8\sqrt{2}\).
- Conclusion
The solutions to the system of equations are the points \((8\sqrt{2}, 64)\) and \((-8\sqrt{2}, 64)\). Since the physical context of the problem implies that the side length of the sun shade must be a positive value, we take the positive solution for \(x\), which is \(8\sqrt{2}\). Thus, the length of the leg of the sun shade is \(8\sqrt{2}\) feet.
Examples
In architecture and engineering, sun shades are often designed with parabolic curves to maximize shade while maintaining aesthetic appeal. If an engineer models the curve of a sun shade with the equation \(y = \frac{1}{2}x^2\), they can determine the span of the shade at a specific height. For instance, at a height of 64 feet, finding the value of \(x\) helps identify the width of the structure. This demonstrates how systems of equations are essential for calculating dimensions in construction and design.